The sum of two numbers is 30 and their difference is 8. 1.Find the numbers by setting up
simultaneous equations and 2.solving by matrix methods.

To solve this problem, we need to find two numbers, \( x \) and \( y \), given the following conditions:

1. The sum of the two numbers is 30: 
   \[
   x + y = 30
   \]
2. The difference between the two numbers is 8:
   \[
   x - y = 8
   \]

### Step 1: Set up the simultaneous equations
We have the following system of equations:

\[
\begin{aligned}
1) & \quad x + y = 30 \\
2) & \quad x - y = 8
\end{aligned}
\]

### Step 2: Write the system as a matrix equation
The system of equations can be written in matrix form as:

\[
\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
30 \\
8
\end{pmatrix}
\]

### Step 3: Solve the system using matrix methods
We solve the matrix equation \( A \mathbf{x} = \mathbf{b} \), where

\[
A = \begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix}
x \\
y
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix}
30 \\
8
\end{pmatrix}
\]

To solve for \( \mathbf{x} \), we need to find the inverse of matrix \( A \) and multiply it by vector \( \mathbf{b} \).

1. **Find the determinant of \( A \):**

\[
\text{det}(A) = (1)(-1) - (1)(1) = -1 - 1 = -2
\]

2. **Find the inverse of matrix \( A \):**

\[
A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix}
-1 & -1 \\
-1 & 1
\end{pmatrix}
= \frac{1}{-2} \begin{pmatrix}
-1 & -1 \\
-1 & 1
\end{pmatrix}
= \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{2}
\end{pmatrix}
\]

3. **Multiply \( A^{-1} \) by \( \mathbf{b} \):**

\[
\mathbf{x} = A^{-1} \mathbf{b} = \begin{pmatrix}
\frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & -\frac{1}{2}
\end{pmatrix}
\begin{pmatrix}
30 \\
8
\end{pmatrix}
\]

\[
\mathbf{x} = \begin{pmatrix}
\frac{1}{2}(30) + \frac{1}{2}(8) \\
\frac{1}{2}(30) - \frac{1}{2}(8)
\end{pmatrix}
= \begin{pmatrix}
15 + 4 \\
15 - 4
\end{pmatrix}
= \begin{pmatrix}
19 \\
11
\end{pmatrix}
\]

### Final Answer:
The two numbers are \( 19 \) and \( 11 \).

Would you like more details on any of the steps, or do you have any questions?

Here are 5 related questions you might find useful:
1. How can you verify the solution of a system of equations using substitution?
2. What are other methods to solve simultaneous equations besides matrix methods?
3. How does the determinant of a matrix affect its invertibility?
4. What happens if the determinant of a matrix is zero when solving linear equations?
5. Can you solve a system of equations graphically, and how does it compare to algebraic methods?

**Tip:** Always check the determinant before attempting to find the inverse of a matrix. If the determinant is zero, the matrix does not have an inverse, and the system may have no unique solution or infinitely many solutions.

The sum of two numbers is 30 and their difference is 8. Find the numbers by setting up simultaneous equations and solving by matrix methods.

Learn how to find two numbers whose sum is 30 and difference is 8 by setting up simultaneous equations and solving them using matrix methods. This problem covers essential algebraic concepts, including the use of determinants and matrix inverses. Suitable for students in Grades 9-12.

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